In this paper, we study the relations between trace inequalities(Sobolev and Moser-Trudinger type), isocapacitary inequalities and the regularity of the complex Hessian and Monge-Ampère equations with respect to a general positive Borel measure. We obtain a quantitative characterization for these relations through properties of the capacity minimizing functions.
证明了X.- J. Wang的Hessian Poincaré不等式以及Trudinger和Wang的Hessian Poincaré不等式是微分几何、共形几何和几何偏微分方程的基础。这些显著的不等式最初是通过梯度流方法建立的。本文给出了迹不等式的直接椭圆证明，并推广到用一般测度代替勒贝格测度的迹不等式。新技术依赖于全球估计的解决方案，以沃尔夫的潜力，和对偶参数利用非交换内积锥上的k-凸函数的海森方程。
In this paper, we obtain some important inequalities of Hessian quotient operators, and global $C^2$ estimates of the Neumann problem of Hessian quotient equations. By the method of continuity, we establish the existence theorem of $k$-admissible solutions of the Neumann problem of Hessian quotient equations.
By using quasi-Banach techniques as key ingredient we prove Poincaré- and Sobolev- type inequalities for $m$-subharmonic functions with finite $(p,m)$-energy. A consequence of the Sobolev type inequality is a partial confirmation of Błocki's integrability conjecture for $m$-subharmonic functions.
We consider convex solutions of nonlinear elliptic equations which satisfy the structure condition of Bian-Guan. We prove a weak Harnack inequality for the eigenvalues of the Hessian of these solutions. This can be viewed as a quantitative version of the constant rank theorem.
A Brunn--Minkowski inequality and Convexity for the 2-Hessian eigenvalue in convex domains
University of Science and Technology of China（中国科学技术大学）；Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）；Università degli Studi di Firenze（佛罗伦萨大学）
This article is devoted to the study of several estimations for a positive solution to a nonlinear weighted parabolic equation on a weighted Riemannian manifold. We therefore derive new Li-Yau type and Hamilton type gradient estimates yielding several consequences. We also derive Hessian estimate and some corollaries for the same equation. Among the applications of our estimates discussed here are Harnack type inequalities, Liouville type theorems and a local time reversed Harnack inequality.
This paper is a survey of some recent results on the validity and the failure of global $W^{2,p}$ regularity properties of smooth solutions of the Poisson equation $Δu = f$ on a complete Riemannian manifold $(M,g)$. We review different methods developed to obtain a-priori $L^p$-Hessian estimates of the form $\| \Hess(u) \|_{L^p} \leq C_1 \| u \|_{L^p} + C_2 \| f \|_{L^p}$ under various geometric conditions on $M$ both in the case of real valued functions and for manifold valued maps. We also present explicit and somewhat implicit counterexamples showing that, in general, this integral inequali…
In this paper, we study the conjecture of Gardner and Zvavitch from \cite{GZ}, which suggests that the standard Gaussian measure $γ$ enjoys $\frac{1}{n}$-concavity with respect to the Minkowski addition of \textbf{symmetric} convex sets. We prove this fact up to a factor of 2: that is, we show that for symmetric convex $K$ and $L,$ $$ γ(λK+(1-λ)L)^{\frac{1}{2n}}\geq λγ(K)^{\frac{1}{2n}}+(1-λ)γ(L)^{\frac{1}{2n}}. $$ Further, we show that under suitable dimension-free uniform bounds on the Hessian of the potential, the log-concavity of even measures can be strengthened to $p$-concavity, with $p>…
